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Single Idea 8631

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism ]

Full Idea

Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects.

Gist of Idea

Cantor says that maths originates only by abstraction from objects

Source

report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21

Book Ref

Frege,Gottlob: 'The Foundations of Arithmetic (Austin)', ed/tr. Austin,J.L. [Blackwell 1980], p.27

A Reaction

Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'.

The 49 ideas from George Cantor

9145 We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor]
15946 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
9992 The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor]
17831 Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake]
15896 Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
9616 A set is a collection into a whole of distinct objects of our intuition or thought [Cantor]
18098 Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
13444 Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
15505 If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
10701 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
10865 The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
17889 CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
13447 Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
18173 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
10232 Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
13465 Only God is absolutely infinite [Cantor, by Hart,WD]
8631 Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
10863 Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
8715 Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
13454 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
13016 The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
14199 Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
10082 There are infinite sets that are not enumerable [Cantor, by Smith,P]
13483 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
15910 Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
8710 The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
15906 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
15905 Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
17798 Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
14136 A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
15903 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
18251 Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
18174 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
10883 Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
13528 Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
9555 Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
15902 Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
15908 It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
13464 Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
10112 The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
15901 Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
18176 Pure mathematics is pure set theory [Cantor]